There are various ranks which have been assigned to a smooth manifold. The following ranks are two examples of such ranks:

- The maximum number of global independent vector fields which can be defined on the manifold
- The maximum number of independent commuting vector fields on a manifold

In this question, we introduce a geometric rank as follows:

Let $(M,g)$ be a Riemannian manifold with a corresponding curvature tensor $R$.

The geometric rank of $M$, denoted by ${\rm Gr}(M)$ is defined as the maximum number of independent vector fields on $M$ which mutually generate a plane with zero sectional curvature.

**Question:** What is this geometric rank for $S^3$ or $S^7$ with their standard structures (standard smooth structures and standard Riemannian metric)?

corankmight be more for this concept, since (at least if we replace $M$ with a suitable open subset) it is maximal for a flat metric. This local version should be related to the rank of $R$ viewed as an element of $\operatorname{End} \bigwedge^2 TM$. $\endgroup$ – Travis Nov 4 '18 at 17:49